I cringe a little at the term "reincarnation" in the case of JtB because usually in common parlance reincarnation implies that the reincarnated person has already died, whereas in the case of Elijah he was assumed to heaven without dying (like Enoch), and it also implies being born anew in the womb. In some of the Enochic pseudepigrapha, we read that Enoch was raised up to heaven in the flesh but in entering the highest heaven his earthly garments were removed and he donned angelic heavenly robes (= he took angelic form, cf. the "spiritual body" mentioned in 1 Corinthians and the "white robes" given to the "great crowd" in heaven in Revelation 7). Although there is no surviving tradition about Elijah's life in heaven, the Enochic parallel suggests that a similar thing may have been thought to have occurred to Elijah. There is also a key apocalyptic scenario in the Book of Parables of 1 Enoch in which the heavenly Enoch will return as the anointed Son of Man, as the eschatological judge of the world....a text that likely lies behind NT apocalyptic references to the Son of Man. Similarly, there are texts at Qumran (particularly 11QMelch) that construes the priestly Messiah as Melchizedek revivivus (obviously related to the christological Melchizedek references in Hebrews). Since there was a similar messianic scenario about Elijah (as Malachi attests), it would not be unusual that he too would resume his function as prophet at the eschaton...and a clear residue of this idea appears in the the "two witnesses" vision of Revelation, the Apocalypse of Elijah, and other later texts.
So it is unclear whether Elijah would return in the same flesh he ascended to heaven in, or would be re-incarnated in a literal sense (i.e. reclothed in human "garments" as he descends from heaven), or whether he returns in a "spiritual body" to use Paul's term. But the Eastern concept of reincarnation, I think, is foreign to Jewish apocalyptic thinking.
Some scholars lean toward the idea that Enoch and Elijah are the 2 witnesses referred to in Revelation chapter 11.
Actually it is clear from the literary allusions in Revelation 11:6 that the witnesses are Elijah and Moses. Compare the story of the transfiguration in the synoptics (cf. Matthew 17:3-4 parr.), in which they appear together, and the eschatological expectation in Deuteronomy Rabbah 3.17 = Midrash Rabbah 6:88. Although Moses died, there are much evidence of a tradition that he ascended in the spirit to heaven as he died (cf. the Assumption of Moses, its allusions in Jude and the apologists, Josephus, Antiquities 4.326, Acts of Pilate 16:7, and of course the transfiguration stories), just as there were similar traditions about Abraham (cf. the Testament of Abraham, the "Rich Man and Lazarus" story in Luke 16, etc.) and other patriarchs. One function of the transfiguration story also is to establish that Jesus was not Moses redivivus or Elijah redivivus. But you are right in that there were also traditions (outside of Revelation) that the "two witnesses" were Enoch and Elijah, the two prophets who never died (cf. Apocalypse of Elijah 4:7, Tertullian, De Anima 50; Hippolytus, De Antichristus 46.3-4, Comm. in Daniel 4.35.3, Apocalypse of Paul 20, Apocalypse of Peter Eth. 2). The big question is whether the Moses + Elijah pairing is older than the Enoch + Elijah pairing or what relationship the two traditions have.